We prove a combinatorial formula for the Poisson bracket of
two elements of the free Lie algebra on two generators, which has a
particularly nice cocycle form when the two elements are Lie monomials
containing only one y. By relating this cocycle form with the period
polynomials introduced by Eichler-Shimura and Zagier, we completely
describe and classify a set of fundamental relations in Ihara's stable
derivation algebra, generalizing the first few cases of these relations
which he had observed and computed by hand.